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A185943
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Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.
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3
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1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 16, 12, 5, 1, 6, 39, 34, 18, 6, 1, 7, 104, 98, 59, 25, 7, 1, 8, 301, 294, 190, 92, 33, 8, 1, 9, 927, 919, 618, 324, 134, 42, 9, 1, 10, 2983, 2974, 2047, 1128, 510, 186, 52, 10, 1, 11, 9901, 9891, 6908, 3934, 1887, 759, 249, 63, 11, 1
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history;
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OFFSET
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0,2
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LINKS
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FORMULA
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R(n,k,m) = k*Sum_{i=0..n-k} binomial(i+m-1, m-1)*binomial(2*(n-i)-k-1, n-i-1)/(n-i), m = 2, k > 0.
R(n,0,2) = n + 1.
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EXAMPLE
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Array begins
1;
2, 1;
3, 3, 1;
4, 7, 4, 1;
5, 16, 12, 5, 1;
6, 39, 34, 18, 6, 1;
7, 104, 98, 59, 25, 7, 1;
8, 301, 294, 190, 92, 33, 8, 1;
Production matrix begins:
2, 1;
-1, 1, 1;
1, 1, 1, 1;
0, 1, 1, 1, 1;
0, 1, 1, 1, 1, 1;
0, 1, 1, 1, 1, 1, 1;
0, 1, 1, 1, 1, 1, 1, 1;
0, 1, 1, 1, 1, 1, 1, 1, 1;
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MATHEMATICA
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r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 2] := n + 1; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2012, from formula *)
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PROG
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(Sage)
@CachedFunction
def A(n, k):
if n==k: return n+1
return add(A(n-1, j) for j in (0..k))
for n in (0..7) :
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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